Weak coupling and spectral instability for Neumann Laplacians.

We prove an abstract criterion on spectral instability of nonnegative selfadjoint extensions of a symmetric operator and apply this to self-adjoint Neumann Laplacians on bounded Lipschitz domains, intervals, and graphs. Our results can be viewed as variants of the classical weak coupling phenomenon for Schrödinger operators in L^2(\mathbb {R}^n) for n=1,2. [ABSTRACT FROM AUTHOR]